Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Suppose that \(a_{1}, a_{2}, a_{3}, \ldots\) and \(b_{1}, b_{2}, b_{3}, \ldots\) are arithmetic sequences. Let \(d_{n}=a_{n}+c \cdot b_{n}\), for any real number \(c\) and every positive integer \(n\). Show that \(d_{1}, d_{2}, d_{3}, \ldots\) is an arithmetic sequence.

Step-by-Step Solution

Answer

The sequence \(d_n\) is arithmetic with common difference \(d_a + c \cdot d_b\).

Step 1: Understand the problem

We are given two arithmetic sequences, \(a_n\) and \(b_n\), with a sequence \(d_n = a_n + c \cdot b_n\). We need to show that \(d_n\) is also an arithmetic sequence, independent of the value of \(c\).

Step 2: Recall the definition of an arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference.

Step 3: Express formulas for the sequences

Assume \(a_n\) is given as \(a_n = a_1 + (n-1)d_a\) and \(b_n = b_1 + (n-1)d_b\), where \(d_a\) and \(d_b\) are the common differences of the sequences \(a_n\) and \(b_n\), respectively.

Step 4: Substitute into the given expression for \(d_n\)

Substitute the expressions for \(a_n\) and \(b_n\) into \(d_n\):\[d_n = a_1 + (n-1)d_a + c \cdot [b_1 + (n-1)d_b].\] Simplifying gives us:\[d_n = (a_1 + c \cdot b_1) + (n-1)(d_a + c \cdot d_b).\]

Step 5: Identify the common difference for \(d_n\)

The expression for \(d_n\) can be seen as: \((a_1 + c \cdot b_1) + (n-1)(d_a + c \cdot d_b)\). The term \((d_a + c \cdot d_b)\) is constant for all \(n\), hence it is the common difference for the sequence \(d_n\).

Step 6: Conclude that \(d_n\) is an arithmetic sequence

Since the expression for \(d_n\) fits the form of an arithmetic sequence with a constant common difference \(d_a + c \cdot d_b\), we conclude that \(d_1, d_2, d_3, \ldots\) form an arithmetic sequence.

Key Concepts

Sequences

Sequences are fundamental in mathematics, often serving as the building blocks for more complex concepts. A sequence is essentially an ordered list of numbers. Each number in a sequence is called a term and has a specific position, such as first, second, third, and so on. Sequences can be finite, having a specific number of terms, or infinite, continuing indefinitely. They offer a systematic way of analyzing patterns and predicting future terms, which is why they're so integral to various fields of study.

• A sequence is defined by a unique formula that describes the relationship between its terms.
• Each term in the sequence is determined by its position, often denoted by the variable \( n \).
• While there are different types of sequences, arithmetic sequences are among the most straightforward and widely studied in mathematics.

Understanding sequences is crucial because they appear in numerous applications, not just in math, but in areas like finance, computer science, and physics. The predictability of sequences allows for modeling and solving real-world problems efficiently.

Common Difference

The common difference is a critical concept that defines arithmetic sequences. In any arithmetic sequence, the difference between consecutive terms is constant. This consistent interval is what we call the 'common difference'. It helps in both generating terms from a formula and finding unknowns within a sequence.
A simple way to think about the common difference is it’s the 'step' you take to get from one number to the next in the sequence. If you know the common difference, you can predict or calculate any term in the sequence. For instance:

• If the first term \( a_1 \) is known and the common difference \( d \), any term \( a_n \) can be found using \[ a_n = a_1 + (n-1) imes d \].
• The concept of common difference is crucial because it defines the linearity of the sequence, ensuring that each term progresses by adding a fixed number.
• It allows for easy alterations; for instance, in combination with other sequences, as seen in our problem where combining deposits results in a new common difference.

By leveraging the idea of a common difference, arithmetic sequences can be manipulated to suit a variety of purposes, leading to profound insights in both pure and applied mathematics.

Algebraic Expressions

Algebraic expressions are used in sequences to formulate terms and relationships efficiently. An algebraic expression is a mathematical phrase that combines numbers, variables, and operators in some form, such as addition or multiplication, to represent a specific value or relation.
For the given problem, algebraic expressions allow us to define the terms of our arithmetic sequences concisely. We write terms like \( a_n = a_1 + (n-1)d_a \) or \( b_n = b_1 + (n-1)d_b \). Here’s why they matter:

• They simplify complex problems by providing structured representations, making it easier to handle calculations symbolically rather than numerically.
• With algebraic expressions, adjustments and substitutions become straightforward, as seen when terms from \( a_n \) and \( b_n \) are combined into a new sequence \( d_n = a_n + c \, b_n \).
• These expressions help illustrate the properties of the sequences involved. By seeing how terms align algebraically, we deduce whether a new sequence retains properties like being arithmetic.

Algebraic expressions streamline solving and understanding sequence problems, making them a powerful tool in both theoretical and practical scenarios.